Exact values at 210 degrees

Text

Compute the exact values of $\sin 210^\circ$, $\cos 210^\circ$ and $\tan 210^\circ$.

Solution

$210^\circ=180^\circ+30^\circ$: the angle lies in the third quadrant, where sine and cosine are both negative. Using associated arcs with reference angle $30^\circ$: $\sin 210^\circ=-\sin 30^\circ=-\tfrac12,\qquad \cos 210^\circ=-\cos 30^\circ=-\tfrac{\sqrt3}{2}.$ The tangent is their ratio and is positive in the third quadrant: $\tan 210^\circ=\frac{\sin 210^\circ}{\cos 210^\circ}=\frac{-1/2}{-\sqrt3/2}=\frac{1}{\sqrt3}=\frac{\sqrt3}{3}\approx 0.577.$ $\boxed{\,\sin 210^\circ=-\tfrac12,\ \cos 210^\circ=-\tfrac{\sqrt3}{2},\ \tan 210^\circ=\tfrac{\sqrt3}{3}\,}$